Download - Introduction to Wavelets
David HerrinUniversity of Kentucky
Introduction to Wavelets
Vibro-Acoustics Consortium
July 29, 2021
Vibro-Acoustics Consortium
July 29, 2021
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What is a Wavelet?
A waveform of limited duration with an average value of zero (i.e. a small wave).
Haar Wavelet Morlet Wavelet Daubechies 2 (db2)
Vibro-Acoustics Consortium
July 29, 2021
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What is a Wavelet?
Stretching the Haar Wavelet
+1 -1+1 +1 -1 -1+1 +1 +1 +1 -1 -1 -1 -1+1 +1 +1 +1 +1 +1 +1 +1 -1 -1 -1 -1 -1 -1 -1 -1
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What is a Wavelet?
Time
Frequency
Precision
Time
Frequency
Precision
Heisenberg Uncertainty Principle – It is impossible to know exact frequency content at an exact time.
Same Area
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July 29, 2021
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Wavelet Transform
Time
Frequency
Precision
Intuitively
Time
Scale
Precision
Wavelet
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Music Analogy (i.e. Scale)
Low notes (low frequency) need longer to be correctly generated while high notes (high frequency) can be generated quickly.
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Wavelet Transforms
• Continuous Wavelet Transform • Discrete Wavelet Transform
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Continuous Wavelet Transform Step 1
Take a wavelet and compare it to a section at the start of the original signal.
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Calculate a number 𝐶 that indicates how closely correlated the wavelet is with this section of the signal. The higher 𝐶 is the more similarity. Correlation depends partly on the wavelet selected.
Continuous Wavelet Transform Step 2
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Shift wavelet slightly to the right and repeat steps 1 and 2.
Continuous Wavelet Transform Step 3
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Scale (stretch) the wavelet slightly and repeat again.
Continuous Wavelet Transform Step 4
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Repeat the process for all scales. If we do this at 10 scales and for 400 time steps, we will have a 400 by 10 set of correlation coefficients. A wavelet that resembles what you are looking for will give a strong correlation at some stretch and shift.
Continuous Wavelet Transform Step 5
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Continuous Wavelet Transform
Wavelet Signal
Translation (Position) Parameter
Scale (Dilation) Parameter
Time SignalCorrelation Coeffient
𝐶 𝑎, 𝑏1𝑎
𝑓 𝑡 𝜓𝑡 𝑏𝑎 𝑑𝑡
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Example Non-Stationary Signal
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Example Non-Stationary Signal
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Example Cab Reverberation
Balloon located close to door towards front of the passenger compartment.
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107 Hz
196 Hz
Example Cab Reverberation
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Continuous Wavelet Transform
Smooth shifting and scaling
Use any wavelet you want
Inverse CWT is difficult
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Discrete Wavelet Transform
Signals can be reconstructed after being DWT
Scales are powers of 2
Wavelets must be constructed from digital filters
Daubechies 2 (db2)
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July 29, 2021
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Wavelet FiltersLow Pass Filter Low Pass Reconstruction Filter
High Pass Filter High Pass Reconstruction Filter
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Wavelet Filters
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low-pass
high-pass
Approximation 1000 Samples
Approximation and Details
1000 Samples
Details 1000 Samples
S
D
A
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July 29, 2021
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Multiple Level Decomposition
S
cA1 cD1
cA2 cD2
cA3 cD3
Frequency
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Downsampling and Upsampling
0.12 0.15 0.18 0.15 0.12 0.09 0.06 0.03 0.00 0.03 0.05 0.09 0.12
Original Signal
Downsampled Signal (remove every other data point)
0.12 0.18 0.12 0.06 0.00 0.05 0.12
Upsampled Signal (insert 0’s between every other data point)
0.12 0.00 0.18 0.00 0.12 0.00 0.06 0.00 0.00 0.00 0.05 0.00 0.12
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low-pass
high-pass
Approximation 500 Samples
With Wavelet Filters
1000 Samples
Details 500 Samples
S
cD
cA
Downsample
Downsample
Downsample – Throw away every other term in the data
A
D
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Discrete Wavelet Transform
S
𝐿
𝐻
𝐿
𝐻
S
cA
cD
cA
cD
A
D
A
D
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Wavelet Transforms
• Continuous Wavelet Transform • Discrete Wavelet Transform
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D. L. Fugal, Conceptual Wavelets in Digital Signal Processing: an In-Depth, Practical Approach for the Non-Mathematician, Space and Signals Technical Publishing, San Diego (2009).
Wavelets Reference